Friday, May 18, 2007

Something is bothering me at the boundary of physics and metaphysics. It seems very likely that a successful theory of quantum gravity will entail that our actual universe is finite. This follows from two considerations. First, in the new theory, the singularities of general relativity will be banished, and the universe will be seen to be grained at the Planck scale. Second, it seems to me that the observable universe can be identified with the actual universe: in what sense should we consider a putative region of the universe beyond the reach of any possible causal contact to be actual? So, it follows that the actual universe is finite.

Now in reading metaphysical papers recently by Ross Cameron and Jonathan Schaffer, (see posts here and here), I was introduced to the argument for the conceivability of “gunk”. Gunk is stuff every part of which has proper parts -- that is, it is infinitely divisible. Now is a world made of gunk conceivable? It seems so. Now, since I have embraced the general stance that conceivability implies possibility, I would have to concede that if the actual world is finite, this is a contingent rather than necessary fact about the world.

For some reason, this just rubs me the wrong way. I don’t like thinking that something as fundamental as the conclusion that our world is finite in extent is just a contingent fact. But given that we are (famously) adept at conceiving infinities, and the strength of my opinion regarding the modal rationalist link between conceivability and possibility, I’m stuck.

The only strategy which I think might work is as follows. I could assert that the conceivability of infinity is grounded by the whole space of possible worlds, and its application to a single possible world is a mistake. The gunky world would itself have to comprise all possible worlds by virtue of its infinite extent. It would itself necessarily constitute the entire modal space, so it couldn’t also be one of the constituents of modal space. Individual possible worlds themselves would be necessarily finite in this scheme.

Tuesday, May 15, 2007

So, how did this party get started? In Programming the Universe (see also priorposts on this topic), Seth Lloyd would like to retell the cosmological story with qubits instead of elementary particles. However, the section of the book (chapter 3) where he does this doesn’t really add much to the standard account. He interprets fluctuations in quantum fields as superpositions of bits whose possible outcomes "0" and "1" represent low and high energy density. The collapse (or decoherence, following Lloyd’s preferred interpretation) of these superpositions creates pockets of high density which can then be the target of gravitational attraction. If you take out the references to bits, his story seems to be the standard cosmological model. If the universe is really a quantum computer, then matter-energy fields (and space-time) would be derived from qubits.

In other words, the real innovation would come if the computational model helps point us toward a theory of quantum gravity. And here, Lloyd does have some ideas. The book has just a few pages on this, but more detail is found in his paper, “A theory of quantum gravity based on quantum computation”. Some impressions from this paper are below (with the caveat that as usual I can’t understand large portions of it).

The idea is that the metric structure of space-time and the behavior of quantum matter fields “are derived from and arise out of an underlying quantum computer. (p.2)”. One starts with the fact a quantum computer can be thought of as a universal theory for discrete quantum mechanics. Quantum computers represent a causal network (=computational history) of interactions – actually superpositions of such networks. These can be represented as a graph, similar to those in causal set theory. Now, for the matter side of things, note that at each vertex of the graph (=logic gate), qubits can be transformed or not. When they are transformed, this is a scattering event. Each computation is a superposition of different computational histories, one for each pattern of scattering events. The events are the matter.

On the gravity side of things, the superpositions of these computational histories will be seen to correspond to a fluctuation of space-time geometry. Lloyd’s strategy is to “embed the computational graph in a space-time manifold by mapping [the computational graph] C into R4 via an embedding mapping E. (pp.6-7)”. He says that if you do this, then general covariance will follow from the fact that the informational flow through the network is independent of the way the computation is embedded in space-time. The next step (which seems to be the key part of the paper) makes some additional assumptions so that the geometries derived from the computation explicitly obey the Einstein equations (in their discrete Regge calculus form).

Now I can’t follow all the steps here, but what I think he is doing amounts to a demonstration of how a quantum computation could be consistent with the emergence of general relativistic space-time, rather than showing that it would actually do so as a matter of course. He ends up being at least partially circular in invoking our knowledge of the Einstein equations to achieve his explicit results (if someone would like to correct me on this, please do). In contrast, Fotini Markopoulou’s desired ambition (see here and here) is to show that the emergence of space-time is a general consequence of an underlying quantum micro-theory (likewise Olaf Dreyer).

The paper finishes with some ideas on how such a theory would impact a variety of topics in cosmology. For instance, singularities correspond to bits entering or leaving the universe, and black holes do lose information; the model can handle different stages of cosmological evolution, etc. This is interesting stuff, and I’ll be interested in seeing if these ideas are developed further.

Something which intrigues me is how one is supposed to think about this new proposed atom of the universe, the qubit. A practical quantum computer uses properties of familiar particles (spin of an electron or polarization of a photon) as qubits. But if these particles (as well as space-time itself) are derived from these postulated elementary qubits, what are they? Is the superposed atomic qubit just a pure possibility of existence?

[UPDATE: 25 May, 2007. My comments in first paragraph of this post are a bit unfair since later in the book (Ch.8 p.196) Lloyd revisits the story of the history of the universe incorporating some of the ideas from his sections on quantum gravity and complexity. In this discussion, here the computation does indeed have priority status over matter and gravity.]